Carlson's Fractal Gallery

3D Phoenix Spirals

The images in this group were all created
using
Clifford
Pickover's quartic variation of Ushikis's "Phoenix" Julia set
equations:
Z
= Z*Z - .5Z + C, X = Z*Z - .5Y + C, Y = Z, Z = X (see "The World of
Chaos" by Clifford A. Pickover, "Computers in Physics" Sep/Oct 1990).
For
these images, C = (0.563, 0.0) was used instead of C =
(0.56667,
0.0) as given in Pickover's paper. The image was colored using the atan
method (see "Atan Method Fractals," below). Here's a few images (480 x
640 x 256 colors) in the style of the one above selected from my very
large
collection:

Atan Method Fractals

The images in this group were created using
a
variety of equations, most involving transcendental functions. The
colormap
for these images consisted of two color ranges, each range varying in
intensity
linearly from light to dark. Which range was used for a particular
pixel
depended on whether the number of iterations at bailout was odd or
even.
The angle formed by a line joining the last two points in the orbit and
the real axis was computed. The absolute value of this angle was
converted
to a colormap index in the proper color range, and the pixel was
plotted
using that color. Here's a few images (480 x 640 x 256 colors) in the
style
of the one above selected from my very large collection:

3D Stalks

The images in this group were created using
a
variation of a method developed by Clifford
Pickover in which pixels are plotted in certain colors if the
absolute
value of either the real or imaginary component of Z falls
below
a specified value. In my variation of this method, this absolute value
was converted to an index into one of two color ranges in the colormap,
depending on whether the number of iterations was odd or even. A wide
variety
of equations were used to produce these images. Here's a few images
(480
x 640 x 256 colors) in the style of the one above selected from my very
large collection:

Bubbles

The images in this group were all created
using
the standard Mandelbrot equation Z = Z*Z + C. The images were
created
in two passes. On the first pass, the minimum value of the modulus of Z
obtained in the iteration loop for each pixel was converted to a
colormap
index, and the pixel was plotted using that color. When the first pass
was complete the program would pause until I clicked the mouse with the
cursor over a pixel, say pixel A, in the image. The minimum
modulus
of Z for pixel A was computed and then a second pass
was
made in which all pixels having a minimum modulus of Z less
than
that of pixel A's were plotted the background color. Here's a
few
images (480 x 640 x 256 colors) in the style of the one above selected
from my very large collection:

Pokorny Cones

The images in this group were all created
using
the Pokorny equation Z = 1 / (Z*Z + C). The pixels were colored
using the atan method (see "Atan Method Fractals," above). This method
gives many of the images the appearance of consisting of an infinite
number
of (sometimes quite distorted) cones. Here's a few images (480 x 640 x
256 colors) in the style of the one above selected from my very large
collection:

Miscellaneous Images

The fractals in this section were produced
using
a variety of equations and rendering methods. Many of the fractals were
rendered by methods utilizing one or more "orbit traps." An orbit trap
is a bounded area in the complex plane of some simple shape such as a
circle
or square. When a point in the iteration orbit falls inside an orbit
trap,
the iteration loop is exited and the distance from the point to the
center
of the trap is used to index into two or more color ranges of the
colormap.

Many people continue to ask me how I created a particular fractal.
The answer is, that all the fractals I post were created with programs
that I wrote myself over the past fifteen years. These programs were
not designed to be used by anyone but myself and were never made
available on the web.

However, I have recently completed a program designed especially
to create my style of fractals, fractals with sharply defined elements
having a rounded 3D appearance. The program is designed to be
very easy to use, especially for beginners and requires no knowledge
of the mathematics involved.